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algebra, a Weyl module is a representation of a reductive algebraic group, introduced by Carter and Lusztig (1974, 1974b) and named after Hermann Weyl. In characteristic 0
Weyl_module
Mathematical theorem in representation theory
Schur–Weyl duality is the statement that the space of two-tensors decomposes into symmetric and antisymmetric parts, each of which is an irreducible module
Schur–Weyl_duality
Module over a sheaf of differential operators
the Weyl algebra to differential equations. An (algebraic) D-module is, by definition, a left module over the ring An(K). Examples for D-modules include
D-module
Non-tensorial representation of the spin group
3-dimensional Euclidean space are quaternionic, Weyl spinors in 4-dimensional Euclidean space are quaternionic, Weyl spinors in Lorentzian signature ( 3 , 1 )
Spinor
Objects in representation theory of Lie algebras
representations of Lie algebras Theorem of the highest weight Generalized Verma module Weyl module E.g., Hall 2015 Chapter 9 Hall 2015 Section 9.2 Hall 2015 Sections
Verma_module
Weyl group Weyl integral Weyl integration formula Weyl law Weyl metrics Weyl module Weyl notation Weyl quantization Weyl relations Weyl scalar Weyl semimetal
List of things named after Hermann Weyl
List_of_things_named_after_Hermann_Weyl
a Jantzen filtration is a filtration of a Verma module of a semisimple Lie algebra, or a Weyl module of a reductive algebraic group of positive characteristic
Jantzen_filtration
Differential algebra
algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced
Weyl_algebra
Relativistic wave equation describing massless fermions
inherent handedness, or chirality, called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types
Weyl_equation
Direct sum of irreducible modules
in which they are described as nonartinian simple rings. The module theory for the Weyl algebras is well studied and differs significantly from that of
Semisimple_module
Concept in Lie algebra representation theory
it is convenient to choose an inner product that is invariant under the Weyl group, that is, under reflections about the hyperplanes orthogonal to the
Weight (representation theory)
Weight_(representation_theory)
the characters of Demazure modules, and is a generalization of the Weyl character formula. The dimension of a Demazure module is a polynomial in the highest
Demazure_module
matrices or 8×8 real matrices are needed. Weyl–Brauer matrices Higher-dimensional gamma matrices Clifford module bundle Atiyah, Michael F.; Bott, Raoul;
Clifford_module
product Schur product theorem Schur test Schur's property Schur's theorem Schur's number Schur–Horn theorem Schur–Weyl duality Schur–Zassenhaus theorem
List of things named after Issai Schur
List_of_things_named_after_Issai_Schur
was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural
Representation theory of semisimple Lie algebras
Representation_theory_of_semisimple_Lie_algebras
case of the splitting of a module for a semisimple Lie group into its irreducible factors. In dimension 4, the Weyl module decomposes further into a pair
Ricci_decomposition
Integral polynomial
finite Weyl group. For each w ∈ W denote by Mw be the Verma module of highest weight −w(ρ) − ρ where ρ is the half-sum of positive roots (or Weyl vector)
Kazhdan–Lusztig_polynomial
Mathematical formula
In mathematics, the Weyl integration formula, introduced by Hermann Weyl, is an integration formula for a compact connected Lie group G in terms of a maximal
Weyl_integration_formula
finite-dimensional module over g {\displaystyle {\mathfrak {g}}} is semisimple as a module (i.e., a direct sum of simple modules.) Weyl's theorem implies
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
Operation that pairs a left and a right R-module into an abelian group
of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction
Tensor_product_of_modules
(In the positive characteristic case, the construction only produces Weyl modules, which may not be irreducible.) branching branching rule Brauer Brauer's
Glossary of representation theory
Glossary_of_representation_theory
German mathematician (1882–1935)
described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics
Emmy_Noether
Device in the representation theory of Lie groups
introduced by Adolf Hurwitz (1897) for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation
Unitarian_trick
American mathematician
supervision of Warren J. Wong with dissertation The Submodule Structure of Weyl Modules for Groups of Type An. After post-doctoral positions at University of
Stephen_R._Doty
Associative algebra introduced by Richard Brauer
does for the representation theory of the general linear group in Schur–Weyl duality. The Brauer algebra B n ( δ ) {\displaystyle {\mathfrak {B}}_{n}(\delta
Brauer_algebra
}}} where ⋅ {\displaystyle \cdot } is the affine action of the Weyl group. The Verma module M λ {\displaystyle M_{\lambda }} is called singular, if there
Generalized_Verma_module
Type of Kac–Moody algebras
T} in the vertex algebra. The Weyl group of an affine Lie algebra can be written as a semi-direct product of the Weyl group of the zero-mode algebra
Affine_Lie_algebra
Basic result in the representation theory of Lie groups
first need to describe the Weyl group action centered at − ρ {\displaystyle -\rho } . For any integral weight λ and w in the Weyl group W, we set w ∗ λ :=
Borel–Weil–Bott_theorem
Representation theory of groups
suitable space of functions on G, with G acting by translation. See Peter–Weyl theorem for the compact case. If G is a Lie group but not compact nor abelian
Regular_representation
Isomorphism of commutative rings constructed in the theory of Lie algebras
subalgebra h {\displaystyle {\mathfrak {h}}} that are invariant under the Weyl group W {\displaystyle W} . Let g {\displaystyle {\mathfrak {g}}} be a semisimple
Harish-Chandra_isomorphism
Monster and modular connection
known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky
Monstrous_moonshine
the Verma module with highest weight λ {\displaystyle \lambda } . If λ {\displaystyle \lambda } is sufficiently far inside the fundamental Weyl chamber
Kostant_partition_function
Mathematical property
theory, "semi-simplicity" is also called complete reducibility. For example, Weyl's theorem on complete reducibility says a finite-dimensional representation
Semi-simplicity
ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic
Primitive_ring
is r. Given a right module M over the Weyl algebra A n {\displaystyle A_{n}} , the Gelfand–Kirillov dimension of M over the Weyl algebra coincides with
Gelfand–Kirillov_dimension
nonassociative algebras. An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with multiplication-by-scalars
List_of_algebras
irreducible representations of GLn. In that case, one can consider the Weyl modules associated to a partition λ, which can be described in terms of bideterminants
Garnir_relations
Topological group with compact topology
construction using Verma modules. In Weyl's approach, the construction is based on the Peter–Weyl theorem and an analytic proof of the Weyl character formula
Compact_group
Pictorial representation of symmetry
Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various
Dynkin_diagram
Branch of mathematics that studies abstract algebraic structures
their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation
Representation_theory
Representation theory of the symplectic group
unitary groups of operators, largely through the contributions of Hermann Weyl, Marshall Stone and John von Neumann. In turn these results in mathematical
Oscillator_representation
the PBW theorem. Similarly, a Weyl algebra is almost commutative. Ore condition Gelfand–Kirillov dimension Victor Ginzburg, Lectures on D-modules v t e
Almost_commutative_ring
Mathematical ring with well-behaved ideals
modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of indecomposable injective modules
Noetherian_ring
British mathematician and computer scientist
completing a PhD in 1992. Her doctoral dissertation, Integral Forms for Weyl Modules of G L ( 2 , Q ) {\displaystyle \mathrm {GL} (2,\mathrm {Q} )} , was
Perdita_Stevens
Mathematical operation on vector spaces
include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the universal enveloping algebra in general. The exterior algebra
Tensor_product
Writing Lie algebra sets as matrices
of a Lie group Weight (representation theory) Weyl's theorem on complete reducibility Root system Weyl character formula Representation theory of a connected
Lie_algebra_representation
Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate
Schur_algebra
Direct sum of simple Lie algebras
-modules, the result known as the theorem of the highest weight. The character of a finite-dimensional simple module in turns is computed by the Weyl character
Semisimple_Lie_algebra
Algebraic construct of interest in theoretical physics
representation is invariant under the Weyl group for G, and the representation is integrable. Conversely, if a highest weight module is integrable, then its highest
Quantum_group
Mathematical concept
mathematics, an algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the
Algebraic_character
Wang, Jian Pan (1982), "Sheaf cohomology on G/B and tensor products of Weyl modules", Journal of Algebra, 77 (1): 162–185, doi:10.1016/0021-8693(82)90284-8
Good_filtration
Theoretical object in mathematics
from other ideas of groups over F1, in that the F1‑scheme is not itself the Weyl group of its base extension to normal schemes. Lorscheid first defines the
Field_with_one_element
Branch of mathematics
ideals. For instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring. Therefore,
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1. For groups over finite fields, these
Steinberg_representation
Ring that is also a vector space or a module
and scalar multiplication operations together give A the structure of a module or vector space over K. In this article we will also use the term K-algebra
Associative_algebra
Group representation
always a unique Weyl group element w 0 {\displaystyle w_{0}} mapping the negative of the fundamental Weyl chamber to the fundamental Weyl chamber. Then
Dual_representation
Gamma matrices for arbitrary Clifford algebras
with period 8. (cf. the Clifford algebra clock.) Weyl–Brauer matrices Dirac spinor Clifford module It is possible and even likely that many or most of
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
Concept in ring theory and homological algebra
example Z {\displaystyle \mathbb {Z} } has global dimension one. The first Weyl algebra A1 is a noncommutative Noetherian domain of global dimension one
Global_dimension
German mathematician (1862–1943)
remained there for the rest of his life. Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John
David_Hilbert
Algebraic structure
1 ≠ 3 x 1 x 2 {\displaystyle 2x_{1}x_{2}+x_{2}x_{1}\neq 3x_{1}x_{2}} The Weyl algebra A n ( C ) {\displaystyle A_{n}(\mathbb {C} )} , being the ring of
Noncommutative_ring
Style sheet language
from the original on 2 June 2015. Retrieved 2 June 2015. Meyer, Eric A.; Weyl, Estelle (2023). Cascading Style Sheets: The Definitive Guide, Fifth Edition
CSS
Set of vectors used to define coordinates
every module has a basis. A module that has a basis is called a free module. Free modules play a fundamental role in module theory, as they may be used
Basis_(linear_algebra)
American computer scientist
Animations Level 1". www.w3.org. "CSS Overflow Module Level 3". www.w3.org. "CSS Transitions". www.w3.org. Weyl, Estelle (April 14, 2016). Transitions and
David Baron (computer scientist)
David_Baron_(computer_scientist)
Anti-particle to the electron
electrons having either positive or negative energy as solutions. Hermann Weyl then published a paper discussing the mathematical implications of the negative
Positron
"Smallest" commutative algebra that contains a vector space
these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring. It is possible to
Symmetric_algebra
Representation of the symmetry group of spacetime in special relativity
Clifford algebra and its spin group on a spinor module. These expressions for Dirac spinors and Weyl spinors all extend by linearity of Lie algebras and
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Algebraic structure with addition and multiplication
the order of G (Maschke's theorem). Clifford algebras are semisimple. The Weyl algebra over a field is a simple ring, but it is not semisimple. The same
Ring_(mathematics)
toral subalgebra Unitarian trick Verma module Weyl 1. Hermann Weyl (1885 – 1955), a German mathematician 2. A Weyl chamber is one of the connected components
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Deformation of the group algebra of a Coxeter group
Hecke algebras in general. The case leading to the Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk
Iwahori–Hecke_algebra
Type of ring in non-commutative algebra
zero, the Weyl algebra is simple but not semisimple, and in particular not a matrix algebra over a division algebra over its center; the Weyl algebra is
Simple_ring
Cohomology theory for Lie algebras
results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem. Let g {\displaystyle {\mathfrak
Lie_algebra_cohomology
Associative algebra generalizing the Virasoro algebra
simple roots are sums of any number of consecutive simple roots, and the Weyl vector is their half-sum ρ = 1 2 ∑ e > 0 e {\displaystyle \rho ={\frac {1}{2}}\sum
W-algebra
Theorem in representation theory
paper. The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple
Theorem_of_the_highest_weight
Group of symmetries of an n-dimensional hypercube
{\displaystyle \mathrm {FI} _{\mathcal {W}}} -modules and stability criteria for representations of classical Weyl groups", Journal of Algebra, 420: 269–332
Hyperoctahedral_group
semisimple Yetter-Drinfeld module, Amer. J. Math. 132 (2010), no. 6, 1493–1547 Cuntz: Crystallographic arrangements: Weyl groupoids and simplicial arrangements
Nichols_algebra
Indian mathematician (1933-2012)
ordering on a Weyl Group, Ann. Sci. Ecole Norm. Sup. 4e Serie, t.4, pp. 393–398. J.E.Humphreys and D.N.Verma (1973), Projective Modules for finite Chevalley
Daya-Nand_Verma
Certain functors from the category of modules over a fixed commutative ring to itself
any (possibly negative) integer. In this context Schur-Weyl duality states that as a GL(V)-module V ⊗ n = ⨁ λ ⊢ n : ℓ ( λ ) ≤ k ( S λ V ) ⊕ f λ {\displaystyle
Schur_functor
Finite dimensional algebra over a field whose central elements are that field
to the definition: for instance, for a field F of characteristic 0, the Weyl algebra F [ X , ∂ X ] {\displaystyle F[X,\partial _{X}]} is a simple algebra
Central_simple_algebra
Multi particle state space
representation of the canonical commutation relations, or equivalently of the Weyl algebra. For fermions, the antisymmetric Fock space gives the corresponding
Fock_space
Algebraic structure
independents. A module that has a basis is called a free module, and a submodule of a free module needs not to be free. A module of finite type is a module that
Commutative_ring
Concept in mathematics
the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the Weyl character formula
Reductive_group
Duality for Galois modules for the absolute Galois group of a non-archimedean local field
representation to refer to more general Galois modules. Rubin 2000, Theorem 1.4.1 Rubin, Karl (2000), Euler systems, Hermann Weyl Lectures, Annals of Mathematics Studies
Local_Tate_duality
Group of unitary matrices
\operatorname {U} (1)\rightarrow \operatorname {U} (n)} inducing the inverse. The Weyl group of U ( n ) {\displaystyle \operatorname {U} (n)} is the symmetric
Unitary_group
American rotary machine gun
(2010). The Gun. Simon & Schuster. pp. 116–119. ISBN 978-1-4391-9653-3. Weyl, A. R. (8 March 1957). "Motor-guns—a Flashback to 1914-18". Flight. 71 (2511):
M134_Minigun
Lie algebra with imaginary simple roots
f_{i})=1} . There is a character formula for highest weight modules, similar to the Weyl–Kac character formula for Kac–Moody algebras except that it has
Generalized_Kac–Moody_algebra
Algebraic structure used in analysis
Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used. A Lie
Lie_algebra
Representation of a group or algebra that is a direct sum of simple representations
semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation
Semisimple_representation
Universal construction in multilinear algebra
include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two
Tensor_algebra
Mathematical study of invariants under symmetries
shining armor of algebra, she sprang forth from Cayley's Jovian head. — Weyl (1939b, p.489) Cayley first established invariant theory in his "On the Theory
Invariant_theory
Geometric space with five dimensions
Kaluza-Klein Cosmology. Singapore: World Scientific. ISBN 981-256-661-9. Weyl, Hermann, Raum, Zeit, Materie, 1918. 5 edns. to 1922 ed. with notes by Jūrgen
Five-dimensional_space
semisimple Lie algebras or compact semisimple Lie groups going back to Hermann Weyl include: For a given dominant weight λ, find the weight multiplicities in
Littelmann_path_model
modular forms. The case leading to the Iwahori–Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk
Hecke_algebra_of_a_pair
Algebraic study of differential equations
algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential
Differential_algebra
Matrices similar to diagonal matrices
p-Schatten class is the set of all operators with finite p-Schatten norm. Weyl, von Neumann, and Kuroda, showed the following: For any p > 1 {\displaystyle
Diagonalizable_matrix
Representations of finite groups, particularly on vector spaces
Peter-Weyl Theorem. It is usually presented and proven in harmonic analysis, as it represents one of its central and fundamental statements. The Peter-Weyl
Representation theory of finite groups
Representation_theory_of_finite_groups
1912–1996 Dutch aircraft manufacturer
Fokker." Fokker, A Living History. Retrieved: 19 December 2010. Weyl 1965, pp. 65–67. Weyl 1965, p. 96. "Motor Guns-A flashback to 1914–18." Flight, 8 March
Fokker
Mathematical transformation in physics
algebra Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra Representation
Time-translation_symmetry
(rcb_generator)". Technical Report. Widynski, Bernard (2017). "Middle-Square Weyl Sequence RNG". arXiv:1704.00358 [cs.CR]. Kneusel, Ron (2018). Random Numbers
List of random number generators
List_of_random_number_generators
Group of unitary complex matrices with determinant of 1
rank n − 1 is given by the set of diagonal matrices with determinant 1. The Weyl group of SU(n) is the symmetric group Sn, which is represented by signed
Special_unitary_group
Each semi-simple algebraic group is geometrically reductive
This is because in characteristic 0 the corresponding module has this dimension by the Weyl character formula, and for n large enough that the line
Haboush's_theorem
WEYL MODULE
WEYL MODULE
Surname or Lastname
English
English : topographic name for someone who lived near a spring or stream, Middle English well(e) (Old English well(a)).German : from a short form of the personal names Wallo, Walilo.German : nickname from Middle High German wël ‘round’.
Biblical
well educated; well brought up
Boy/Male
Indian
Well-established, Well-found
Girl/Female
Muslim
Well-established, Well-found
Girl/Female
Muslim
Well-arranged, Well-ordered
Boy/Male
Hindu
Well wisher, Well to do
Girl/Female
Indian
Well-arranged, Well-ordered
Girl/Female
American, Australian, Christian, Danish, Finnish, French, German, Greek, Portuguese, Swedish
Eloquent; Well-spoken; To Talk Well
Surname or Lastname
English
English : variant spelling of Way.Dutch : variant of Wei.
Girl/Female
Tamil
Hitishini | ஹிதீஷீநீ
Well-wisher
Hitishini | ஹிதீஷீநீ
Girl/Female
Indian
Well-established, Well-found
Girl/Female
Tamil
Well wisher
Girl/Female
Gujarati, Hindu, Indian
Well Wisher; Friend; Well-wisher
Girl/Female
African, Arabic, Muslim
Well-ordered; Well-arranged
Boy/Male
Muslim
Well-established, Well-found
Boy/Male
Hungarian
Well.
Girl/Female
Biblical
Well educated, well brought up.
Boy/Male
Irish
Well.
Boy/Male
Tamil
Well born
Boy/Male
Tamil
Hitakrit | ஹிதாகà¯à®°à®¿à®¤Â
Well wisher, Well to do
WEYL MODULE
WEYL MODULE
Girl/Female
Muslim
Inexhaustible, Limitless
Girl/Female
Indian
Life
Boy/Male
Biblical
The right hand.
Surname or Lastname
Irish
Irish : reduced Anglicized form of either of two Gaelic names, Ó DuibhÃn ‘descendant of DuibhÃn’, a byname meaning ‘little black one’, or Ó DaimhÃn ‘descendant of DaimhÃn’, a byname meaning ‘fawn’, ‘little stag’. These are attenuated versions of Ó Dubháin and Ó Damháin, and are the phonetic origin of Anglicizations with an internal v (as opposed to w, as in Dewan, or monosyllabic forms with an o or u) (see Doane).English and French : nickname, of literal or ironic application, from Middle English, Old French devin, divin ‘excellent’, ‘perfect’ (Latin divinus ‘divine’).
Boy/Male
Irish
Surname.
Surname or Lastname
English
English : variant of Seal.North German : metonymic occupational name for a rope maker, from Middle Low German sēl ‘rope’.
Biblical
prudent; good counselor
Boy/Male
Tamil
Reflection
Girl/Female
Latin
Feminine of Darius; a Persian royal name.
Girl/Female
Arabic, Muslim, Pakistani, Urdu
Moon
WEYL MODULE
WEYL MODULE
WEYL MODULE
WEYL MODULE
WEYL MODULE
v. t.
To promote the weal of; to cause to be prosperous.
n.
One who wishes well, or means kindly.
a.
Correctly informed; provided with information; well furnished with authentic knowledge; intelligent.
p. pr. & vb. n.
of Well
a.
Common weal.
a.
Speaking well; speaking with fitness or grace; speaking kindly.
n.
Prosperity; happiness; well-being; weal.
a.
Safe; as, a chip warranted well at a certain day and place.
a.
Good in condition or circumstances; desirable, either in a natural or moral sense; fortunate; convenient; advantageous; happy; as, it is well for the country that the crops did not fail; it is well that the mistake was discovered.
a.
Polite; well-bred; complaisant; courteous.
a.
Prosperous; well.
a.
Being well folded.
a. & adv.
Well.
a.
Well put together; having symmetry of parts.
imp. & p. p.
of Well
a.
Being in health; sound in body; not ailing, diseased, or sick; healthy; as, a well man; the patient is perfectly well.
v. t.
To pour forth, as from a well.
a.
Spoken with propriety; as, well-spoken words.
a.
Balanced or considered with reference to public weal.
n.
The state or condition of being well; welfare; happiness; prosperity; as, virtue is essential to the well-being of men or of society.